List of top Mathematics Questions

Let \( \varphi : \mathbb{R} \to \mathbb{R} \) be the solution of the differential equation \[ 4 \frac{d^2 y}{dx^2} + 16 \frac{dy}{dx} + 25y = 0 \] satisfying \( \varphi(0) = 1 \) and \( \varphi'(0) = -\frac{1}{2} \). Then, the value of \( \lim_{x \to \infty} e^{2x} \varphi(x) \) is equal to ............ (rounded off to two decimal places).

The value of \[ \lim_{n \to \infty} 8n \left( \left( e^{\frac{1}{2n}} - 1 \right) \left( \sin \frac{1}{2n} + \cos \frac{1}{2n} \right) \right) \] is equal to ............... (rounded off to two decimal places).

Let \( f : \mathbb{R} \to \mathbb{R} \) be a twice differentiable function such that \[ f(0) = 4, \, f(1) = -2, \, f(2) = 8, \, f(3) = 2. \] Then, which of the following is/are TRUE?

For \( n \in \mathbb{N} \), consider the set \( U(n) = \{ x \in \mathbb{Z}_n : \gcd(x, n) = 1 \} \) as a group under multiplication modulo \( n \). Then, which of the following is/are TRUE?

Let \( u_1 = (1, 0, 0, -1) \), \( u_2 = (2, 0, 0, -1) \), \( u_3 = (0, 0, 1, -1) \), \( u_4 = (0, 0, 0, 1) \) be elements in the real vector space \( \mathbb{R}^4 \). Then, which of the following is/are TRUE?

Let \( f : \mathbb{R} \to \mathbb{R} \) be a twice differentiable function such that \[ f(0) = 0, \, f'(0) = 2, \, f(1) = -3. \] Then, which of the following is/are TRUE?

For \( n \in \mathbb{N} \), let \[ x_n = \sum_{k=1}^{n} \frac{k}{n^2 + k}. \] Then, which of the following is/are TRUE?

Let \( \varphi : (-1, \infty) \to (0, \infty) \) be the solution of the differential equation \[ \frac{dy}{dx} = 2 y e^x = 2 e^x \sqrt{y}, \] satisfying \( \varphi(0) = 1 \). Then, which of the following is/are TRUE?

If \( M, N, \mu, w : \mathbb{R}^2 \to \mathbb{R} \) are differentiable functions with continuous partial derivatives, satisfying \[ \mu(x, y) M(x, y) \, dx + \mu(x, y) N(x, y) \, dy = dw, \] then which one of the following is TRUE?

Let \( \Omega \) be the bounded region in \( \mathbb{R}^3 \) lying in the first octant \( (x \geq 0, y \geq 0, z \geq 0) \), and bounded by the surfaces \( z = x^2 + y^2 \), \( z = 4 \), \( x = 0 \) and \( y = 0 \). Then, the volume of \( \Omega \) is equal to:

The number of elements in the set \[ \{ x \in \mathbb{R} : 8x^2 + x^4 + x^8 = \cos x \} \] is equal to:

Let \( x_1 = 1 \). For \( n \in \mathbb{N} \), define \[ x_{n+1} = \left( \frac{1}{2} + \frac{\sin^2 n}{n} \right) x_n. \] Then, which one of the following is TRUE?

Let \( x_1 = 1 \). For \( n \in \mathbb{N} \), define \[ x_{n+1} = \left( \frac{1}{2} + \frac{\sin^2 n}{n} \right) x_n. \] Then, which one of the following is TRUE?

Let \( x_1>0 \). For \( n \in \mathbb{N} \), define \[ x_{n+1} = x_n + 4. \] If \[ \lim_{n \to \infty} \left( \frac{1}{x_1 x_2 x_3} + \frac{1}{x_2 x_3 x_4} + \cdots + \frac{1}{x_{n+1} x_{n+2} x_{n+3}} \right) = \frac{1}{24}, \] then the value of \( x_1 \) is equal to:

Let \( f : \mathbb{R}^2 \to \mathbb{R} \) be defined by \[ f(x, y) = e^{y}(x^2 + y^2) \quad \text{for all } (x, y) \in \mathbb{R}^2. \] Then, which one of the following is TRUE?

Let \( \mathbb{R}/\mathbb{Z} \) denote the quotient group, where \( \mathbb{Z} \) is considered as a subgroup of the additive group of real numbers \( \mathbb{R} \). Let \( m \) denote the number of injective (one-one) group homomorphisms from \( \mathbb{Z}_3 \) to \( \mathbb{R}/\mathbb{Z} \) and \( n \) denote the number of group homomorphisms from \( \mathbb{R}/\mathbb{Z} \) to \( \mathbb{Z}_3 \). Then, which one of the following is TRUE?

Let \( f_1, f_2, f_3 \) be nonzero linear transformations from \( \mathbb{R}^4 \) to \( \mathbb{R} \) and \[ \ker(f_1) \subset \ker(f_2) \cap \ker(f_3). \] Let \( T : \mathbb{R}^4 \to \mathbb{R}^3 \) be the linear transformation defined by \[ T(v) = (f_1(v), f_2(v), f_3(v)) \quad \text{for all } v \in \mathbb{R}^4. \] Then, the nullity of \( T \) is equal to:

Let \( f(x) = 10x^2 + e^x - \sin(2x) - \cos x \), \( x \in \mathbb{R} \). The number of points at which the function \( f \) has a local minimum is:

Let \( \varphi : \mathbb{R} \to \mathbb{R} \) be the solution of the differential equation \[ x \frac{dy}{dx} = (y - 1)(y - 3), \] satisfying \( \varphi(0) = 2 \). Then, which one of the following is TRUE?

Let \( \varphi : (0, \infty) \to \mathbb{R} \) be the solution of the differential equation \[ x \frac{dy}{dx} = \left( \ln y - \ln x \right) y, \] satisfying \( \varphi(1) = e^2 \). Then, the value of \( \varphi(2) \) is equal to:

Let \( C \) denote the family of curves described by \( yx^2 = \lambda \), for \( \lambda \in (0, \infty) \) and lying in the first quadrant of the \( xy \)-plane. Let \( O \) denote the family of orthogonal trajectories of \( C \). Which one of the following curves is a member of \( O \), and passes through the point \( (2, 1) \)?

Let \( G \) be a finite abelian group of order 10. Let \( x_0 \) be an element of order 2 in \( G \). If \( X = \{ x \in G : x^3 = x_0 \} \), then which one of the following is TRUE?

Let \( T : P_2(\mathbb{R}) \to P_2(\mathbb{R}) \) be the linear transformation defined by \[ T(p(x)) = p(x + 1), \quad \text{for all } p(x) \in P_2(\mathbb{R}) \] If \( M \) is the matrix representation of \( T \) with respect to the ordered basis \( \{1, x, x^2\} \) of \( P_2(\mathbb{R}) \), then which one of the following is TRUE?

Define \( T : \mathbb{R}^3 \to \mathbb{R}^3 \) by \[ T(x, y, z) = (x + z, 2x + 3y + 5z, 2y + 2z), \quad \text{for all } (x, y, z) \in \mathbb{R}^3 \] Then, which one of the following is TRUE?

Which one of the following is the general solution of the differential equation \[ \frac{d^2 y}{dx^2} - 8 \frac{dy}{dx} + 16y = 2e^{4x} ? \]